Chapter 1 Points, Lines, Planes, and Angles

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Chapter 1 Points, Lines, Planes, and Angles

• Kyle Place and Bill Swartz

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1-2 Points, Lines, and Planes

• A point is the simplest figure in Geometry. It
  has no size and is represented with a capital
  letter.
• A line is a figure in geometry that extends in
  two directions without ending. It has no
  thickness and is named by two points on it or a
  lowercase cursive letter.

A
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1-2 Continued

• A plane is a four-sided figure that extends in
  all directions and has no thickness. It is
  represented by a capital cursive letter or 3
  noncollinear points.

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1-2 Continued

• Space- the set of all points
• Collinear Points- points all in one line
• Coplanar Points- points all in one plane
• Noncollinear Points- points are not on the same
  line
• Intersection- set of points that are in both
  figures

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1-2 Geometric Hierarchy

• Undefined terms base of everything in geometry,
  has points, lines, plane.
• Definitions are based on undefined terms,
  linear, space, coplanar.
• Postulates Statements that are assumed to be
  true, based on definitions. Ex. Parrallel
  postulate.
• Theorems can be proven true by using postulates,
  definitions, and undefined terms.

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1-3 Segments, Rays and Distance

• Between For N to be between M and P, they must
  be collinear.
• Segment Two points on a line and all the points
  between those points.

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1-3 Continued

• Ray AB and all points c such that B is between A
  and C.
• Opposite Rays two collinear rays that have
  exactly one point in common.
• Ruler Postulate If two points are on the same
  line, they can be assigned number values.

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1-3 Continued

• Segment Addition Postulate If B is between A and
  C, then ABBCAC
• Midpoint a point that divides a segment into two
  congruent segments

AM MB
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Segment Bisector

• Segment Bisector A Line, Ray, Plane, or Segment
  through the midpoint of a segment.

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1-4 Angles

• Angle the figure formed by 2 noncollinear rays
  that share the same endpoint.
• Adjacent Angle Two noncoplanar angles that share
  a common vertex, common ray, and no common
  interior points.

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Angles Continued

• Linear Pair 2 adjacent angles whose noncommon
  sides are opposite rays.
• Angle Bisector Ray that divides an angle into 2
  adjacent congruent angles.

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1-4 Postulates

• Linear Pair Postulate If 2 angles form a linear
  pair then the sum of the measure of their angles
  is 180 degrees.
• Angle Addition Postulate if D is on the interior
  of angle ABC then m of angle ABD m of angle DBC
  m of angle ABC.

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1-5 Theorems and Postulates

• Theorem 1-1 If 2 lines intersect, then they
  intersect in exactly one point.
• Theorem 1-2 Through a line and a point not on
  the line, there is exactly one plane.
• Theorem 1-3 If 2 lines intersect then exactly
  one plane contains them.

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1-5 Continued

• Postulate 5- a line contains at least 2 points, a
  plane contains at least 3 noncollinear points,
  and space contains at least 4 noncollinear
  points.
• Postulate 6- through any 2 points there is
  exactly 1 line.
• Postulate 7- through any 3 points there is at
  least one plane, through any 3 noncollinear
  points there is exactly 1 plane.

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1-5 Postulates Cont.

• Postulate 8- If 2 points are in a plane then the
  line that contains them is in that plane.
• Postulate 9- If 2 planes intersect, they
  intersect in a line.

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WE RULE